Method for ascertaining overvoltages in fuel cells

ABSTRACT

The invention relates to a method for ascertaining the overvoltage of a working electrode in a fuel cell, in which the potential of a reference electrode compared to the grounded counter electrode is measured. For the measurement, a fuel cell comprising a polymer electrolyte membrane is used, in which the counter electrode comprises a lateral edge having at least one convexly curved region, and the electrolyte membrane surface, adjoining the counter electrode, comprises an electrode-free region in which the reference electrode is disposed on the electrolyte membrane surface. In contrast, the working electrode is continuous, which is to say has a large surface. The minimum distance L gap  between the reference electrode and the edge of the counter electrode L gap =3×L l,r  with (a) and (b), where m=ionic conductivity of the electrolyte membrane (Ω −1  cm- 1 ), b ox =Tafel slope of the half cell for the electrochemical reaction of the working electrode l m =membrane layer thickness (cm) and j ox   0 =exchange current density of the catalyst of the working electrode per unit of electrode surface in (A cm −2 ). This arrangement can advantageously be used to ensure that the potential measured at the hydrogen-fed reference electrode corresponds to the overvoltage of the working electrode with sufficient accuracy. The method can be applied to polymer electrolyte membrane fuel cells (PEMFC), to direct methanol fuel cells (DMFC) or to high-temperature fuel cells (SOFC).

The invention relates to a novel fuel cell design, in particular for apolymer electrolyte membrane fuel cell (PEMFC), for ascertainingovervoltages.

BACKGROUND OF THE INVENTION

In an electrochemical cell, the electrochemical reactions at theinvolved electrodes are decisively influenced by the half-cellovervoltages. These overvoltages include both contributions from theactivation and from transport losses of the corresponding half cells. Itis therefore considered to be a primary objective to minimize thesecontributions within the scope of a novel cell concept. One of the mostcommon previous methods for measuring half-cell overvoltages is a methodusing a reference electrode (RE).

A reference electrode is always aimed at measuring the potential Φ ofthe electrolyte membrane at an arbitrary point between two electrodes.Having knowledge of this potential makes it possible, in principle, toseparate the cathode overvoltage and the anode overvoltage. However, ithas been found that suitable positioning of the reference electrode isnot readily apparent.

A typical option for positioning a reference electrode within anelectrochemical cell is to dispose the reference electrode on one cellside, at a certain distance L_(gap) from the edges of the workingelectrode, which are aligned with one another. For this purpose, adesign according to FIG. 1 is proposed, in which the two electrodes orthe lateral edges thereof are aligned with one another.

Within the scope of simulations, B. Adler et al., in J. Electrochem.Soc. 149, E166 (2002), have shown that the distance L_(gap) must begreater than three times the membrane layer thickness l_(m). At thisdistance, the inhomogeneities that are regularly caused by the edge of aworking electrode, and the losses of potential resulting from thehydrogen oxidation reaction (HOR), are negligible, and the potentialmeasured at the reference electrode regularly corresponds to themembrane potential at any arbitrary point along the z-axis between theanode and the cathode in the working area.

The measurements of the membrane potential Φ and of the electrodepotentials then allow the overvoltages at the anode and the cathode tobe separated.

Furthermore, with the aid of reference electrodes it is possible tocarry out electrochemical impedance spectroscopy between the referenceelectrode and any other cell electrode. This technique has already beenapplied in the field of solid oxide fuel cells (SOFC).

In electrochemical cells, in which protons act as charge carriers, thepotential of a hydrogen-fed reference electrode generally corresponds tothe membrane potential Φ at the location of the reference electrode,neglecting the voltage losses due to the hydrogen oxidation/generationreaction.

In the implementation for positioning a reference electrode in a fuelcell according to FIG. 1, however, two essential problems occur. Unlessindicated otherwise, it is assumed hereafter that the anode is grounded,and that all potentials are measured with respect to the anode.

Initially, it shall be noted that even a very small shift (δ) in thealignment of the edges of the working electrodes with respect to thecounter electrode nonetheless causes a large change in the potential atthe reference electrode. This effect has already been sufficientlydiscussed in the literature on solid oxide fuel cells (SOFCs). Severalsuggestions on minimizing this effect have also already been describedthere. An analogous application of this solution to PEMFCs has likewisealready been proposed in the literature, which reports on a system ofworking electrodes precisely aligned by way of laser ablation.

A further problem, which occurs even in electrochemical cells in whichthe edges of the working electrodes are precisely aligned (δ=0), isthat, even though the value of the potential Φ measured by the referenceelectrode corresponds to a point on the z-axis between the workingelectrodes, the exact position of this point on the z-axis is not known.It is therefore not possible, in such a case, to unambiguously evaluateand interpret the measured DC voltage signal of the reference electrode.In this case as well, the separation of the half-cell overvoltages canonly take place by way of impedance spectroscopy.

The latest developments by A. A. Kulikovsky and P. Berg (DE102015001572.9), however, demonstrate that it is possible to determinethe overvoltage of a working electrode in a fuel cell by measuring thepotential of a reference electrode compared to the grounded counterelectrode. A fuel cell according to FIG. 2 is used for this purpose.This includes a large-surface-area (endless) working electrode, which inthis instance is a cathode, and a counter electrode, which has at leastone lateral edge. Within the scope of the invention, an electrode isreferred to as having a large surface area (being endless) when theextension thereof corresponds at least to 10 times the parameter l*, andin particular when the extension of the electrode is greater than themembrane layer thickness by several orders of magnitude. As a result ofthe edge of the counter electrode, an electrode-free region is obtained,compared to the working electrode, on the electrolyte membrane surfaceadjoining the counter electrode, in which the reference electrode isdisposed on the electrolyte membrane surface. Due to the fact that theworking electrode no longer has an edge in the vicinity of the workingarea, there is likewise no longer a problem of the alignment of theelectrode edges.

It was possible to demonstrate that the minimum distance L_(gap) betweenthe reference electrode and the edge of the counter electrode must meetthe following condition:

${{L_{gap} \geq {3 \cdot l_{*}}} = {3 \cdot \sqrt{\frac{\sigma_{m}b_{ox}l_{m}}{2j_{ox}^{0}}}}},$where σ_(m)=ionic conductivity of the electrolyte membrane (Ω⁻¹ cm⁻¹),b_(ox)=Tafel slope of the half cell for the electrochemical reaction ofthe cathode, l_(m)=membrane layer thickness (cm) and j_(ox) ⁰=exchangecurrent density of the catalyst of the cathode per unit of electrodesurface in (A cm⁻²). At distances between the reference electrode andthe counter electrode which correspond at least to this minimumdistance, it can be ensured that the measurement of the potential of thereference electrode substantially corresponds to the overvoltage of theworking electrode, which is to say the cathode.

Using typical cell parameters (see Table 1), this yields a minimumdistance L_(gap) on an order of magnitude of several centimeters, in thecase of the reference electrode being disposed on the anode side forascertainment of the overvoltage on the cathode side. This distance,however, is a very large value for a test fuel cell of normally 10 cm*10cm or smaller.

SUMMARY OF THE INVENTION

It is an object of the invention to provide a method for ascertainingthe overvoltage of a working electrode of a fuel cell which overcomesthe existing disadvantages of the prior art and, in particular, theaforementioned minimum distance from the reference electrode in a fuelcell arrangement. The method should, in particular, be usable forpolymer electrolyte membrane fuel cells (PEMFC), direct methanol fuelcells (DMFC) and high-temperature solid oxide fuel cells (SOFC).

It is furthermore an object of the invention to provide an improved fuelcell arrangement, with the aid of which the aforementioned methodaccording to the invention can be carried out.

The objects of the invention are achieved by a novel fuel cellarrangement according to the main claim and by a method according to theadditional independent claim. Advantageous embodiments of the device andof the method can be found in the respective dependent claims.

Within the scope of the present invention, it was found, supported bynumerical simulations for a polymer electrolyte membrane fuel cell(PEMFC) comprising a concentric small anode and a concentriclarge-surface-area (endless) cathode, that a novel arrangement overcomesthe existing disadvantages of the prior art, based on, and in acontinuation of, the arrangement of a reference electrode in a fuel cellas proposed in A. A. Kulikovsky and P. Berg (DE 102015001572.9). Forthis purpose, a fuel cell arrangement is proposed, in which it ispossible to position a reference electrode at a considerably smallerdistance than before from a counter electrode, and in particular ananode, and it is nonetheless possible to ascertain the overvoltage ofthe counter electrode, and in particular of a cathode, with sufficientaccuracy. Moreover, this novel arrangement of the fuel cell regularlyhas electrode surface areas that are able to ensure sufficientelectrochemical conversion.

A model was developed for a radial distribution of the cathodeovervoltage η_(c) and of the membrane potential Φ in the anode-freeregion of the fuel cell. Mathematically, the problem results in anaxially symmetric Poisson-Boltzmann equation for the cathode overvoltageη_(c). The solution to this problem demonstrates that |η_(c)| shows arapid drop to zero as a function of the radius, while |Φ| quickly risesto the value of |η_(c) ⁰| of the cathode overvoltage in the working areaof the cell. The smaller the radius of the concentric anode R_(a), thefaster the membrane potential Φ reaches the value of the cathodeovervoltage in the working area η_(c) ⁰.

From this, it follows that, in measuring the cathodic overvoltagebetween the (working) electrodes, the smaller the local radius R_(a) ofthis region, the closer the reference cell (RE) can be positioned to theconvexly curved edge region of the curved anode.

Hereafter, the electrode of the fuel cell on which the overvoltage is tobe ascertained is referred to as the working electrode. In contrast, acounter electrode shall be understood to mean a further electrode ofthis fuel cell, which is required for the necessary conduction ofcurrent. Both the working electrode and the counter electrode areelectrodes at which electrochemical processes take place in a controllermanner.

In the literature, model notions of the fuel cells comprising areference electrode are based on the 2D simulations in which themembrane potential Φ is present in the plane perpendicular to theworking electrodes and the reference electrode (x-z plane in FIG. 1).The numerical calculations, however, show neither a dependence nor acharacteristic quantification during variations, for the membranepotential Φ in such systems.

The invention now achieves the object of being able to measure ameaningful overvoltage in a fuel cell by way of a reference electrode,by proposing a fuel cell arrangement comprising an electrolyte membrane,which comprises a counter electrode that is delimited by at least asingle edge and includes at least one convexly curved edge region havinga local small radius of curvature R_(a), and a continuous (endless)working electrode.

Within the scope of the invention, a continuous working electrode shallbe understood to mean an electrode having edges disposed so far awayfrom the working areas, which is to say from the edges of the counterelectrode, that any influence on the working area by the counterelectrode edges can generally be precluded. Again, the term ‘continuousworking electrode’ shall mean that the extension thereof is greater thanthe parameter l* by orders of magnitude.

Adjoining the counter electrode, the fuel cell according to theinvention comprises an electrode-free region on the electrolyte membranesurface. Within this region, the reference electrode is disposed at adistance L_(gap) from the edge of the counter electrode. This distanceis generally a multiple of the membrane thickness. The working electrodeextends on the opposite side of the electrolyte membrane. This islocated opposite both the counter electrode and the reference electrode,and in this regard shall be considered to be continuous. All commonstandard reference electrodes may be used as reference electrodes, forexample a reverse hydrogen electrode (RHE), or a dynamic hydrogenelectrode (DHE), in which an electrolyte potential is measured may beused.

In contrast to the arrangement of A. A. Kulikovsky and P. Berg (DE102015001572.9), which is illustrated in FIG. 3 in a cross-sectionalview (a) and in a top view (b), the counter electrode in the inventiveembodiment of the invention does not show only a straight edge, comparedto the reference electrode, but in the top view has a convex area ofcurvature having a smaller radius of curvature R_(a), at least in theedge region disposed closest to the reference electrode. This is met, ina first approximation, when the anode itself is designed in the mannerof a concentric disk, as is apparent from FIG. 4.

Compared to the previously described straight edges of a counterelectrode according to FIG. 3, the convexly curved region of the counterelectrode having a radius of curvature R_(a) has the advantage that theminimum distance L_(gap), at which the reference electrode is disposedfrom the counter electrode, is considerably smaller than thecorresponding distance L_(gap) in the arrangement according to FIG. 4,while the overvoltage of the working electrode, and in particular of thecathode, can nonetheless be ascertained with sufficient accuracy. Thisis of crucial importance, in particular with small fuel cells.

The reference electrode can advantageously be designed to be very smallwith the arrangement according to the invention. In this way, it ispossible to ensure that the tip of the convexly curved region of thecounter anode is the closest region of the counter electrode. If thiswere not the case, the membrane potential could be impaired at thelocation of the reference electrode by contributions from remote regionsof the anode.

In a first embodiment of the invention (FIG. 5), the counter electrodehas a substantially convex edge, which includes a tip having a radiusR_(a), in the immediate vicinity of which the reference electrode isdisposed at a minimum distance L_(gap).

In this arrangement of the counter electrode, the distance L_(gap) canbe selected to be significantly smaller, which advantageously results ina much more compact design comprising the reference electrode.Nonetheless, such an arrangement advantageously makes it possible thatthe potential measured at the reference cell corresponds with sufficientaccuracy to the overvoltage of the cathode.

During the actual measurement for ascertaining the overvoltage, thereference electrode is fed hydrogen. The counter and working electrodesare fed an operating agent and an oxidizing agent, depending on whetherthey are switched as an anode or cathode. During measurement, thepotential of the reference electrode compared to the grounded counterelectrode is measured.

A PEM fuel cell is usually operated with hydrogen and air or oxygen, aDMFC with methanol or a methanol/water mixture and air or oxygen, and aSOFC with hydrogen and air or oxygen.

In the particular in case where the counter electrode is switched as theanode and the working electrode as the cathode, the measured potentialof the reference cell corresponds to the overvoltage of the cathode withsufficient accuracy.

BRIEF DESCRIPTION OF THE DRAWINGS

Additionally, several select figures are used to further illustrate theinvention; however, these shall not be construed to limit the subjectmatter of the invention. In the drawings:

FIG. 1 shows an exemplary schematic fuel cell arrangement comprising areference electrode (from the prior art);

FIG. 2 shows an exemplary schematic fuel cell arrangement comprising areference electrode (from the prior art);

FIG. 3 shows an exemplary schematic fuel cell arrangement comprising areference electrode (from the prior art) in a side view a) and in a topview b);

FIG. 4 shows a schematic fuel cell arrangement in a top view, comprisinga concentric counter electrode;

FIG. 5 shows a schematic top view onto an embodiment according to theinvention of a fuel cell arrangement comprising an anode having aconvexly curved edge region in close proximity to a reference electrode;

FIG. 6 shows an exemplary schematic fuel cell arrangement comprising aconcentric anode having a radius R_(a) and an (infinite) concentriccathode having a radius R_(c) as a basis for the model calculations;

FIG. 7 shows the curve of the potential of the oxygen reduction reactionas a function of the radial position, standardized for the membranelayer thickness l_(m) for the indicated standardized anode radiusR_(a)/l_(m) according to the model from FIG. 7;

FIG. 8 shows the membrane potential Φ⁺ and the local overvoltage on thecathode side η_(c) ⁺ in the region around the anode edge, in each caseplotted against the standardized radial distance from the edge of theanode (concavely curved edge region), based on the example of a PEMFC;and

FIG. 9 shows the normalized distance from the anode edge to the point atwhich the overvoltages of the oxygen reduction reaction drop to thevalue of the ORR Tafel slope b_(ox) as a function of the standardizedanode radius.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereafter, the fuel cell model and fundamental equations will bedescribed, which serve as the basis for the numerical calculation forthe fuel cell design according to the invention.

The following symbols are used:

˜ denotes non-dimensional variables

b Tafel slope of the half cell for the anodic or cathodic reaction (V)

E^(eq) equilibrium potential of a half cell (V)

F Faraday constant

J average current density in the working area (A cm⁻²)

j_(a) local proton current density on the anode side (A cm⁻²)

j_(c) local proton current density on the cathode side (A cm⁻²)

j_(hy) hydrogen exchange current density (A cm⁻²)

j_(hy) ⁰ hydrogen exchange current density in the working area (A cm⁻²)

j_(ox) oxygen exchange current density (A cm⁻²)

j_(ox) ⁰ oxygen exchange current density in the working area (A cm⁻²)

L_(gap) minimum distance between the edge of the counter electrode andthe edge of the reference electrode

L_(l,x) distance between the straight edge of the counter electrode andthe point at which η_(c) ⁺=b_(ox) (cm)

L_(l,r) radial distance between the edge of the counter electrode andthe point at which _(x)η_(c) ⁺=b_(ox) (cm)

l_(m) membrane layer thickness (cm)

r radial position

R_(a) anode radius

R_(c) cathode radius

z coordinate perpendicular to the membrane surface (cm)

The following subscript indices are used:

a anode

c cathode

HOR hydrogen oxidation reaction

hy hydrogen

m membrane

ORR oxygen reduction reaction

ox oxygen

ref reference electrode

x system comprising a straight edge of the counter electrode

Furthermore, the following superscript indices are used:

+ positive value

0 center of the concentric counter electrode (=0) {tilde over (r)}

∞ for an infinite radius

Additionally, the following Greek symbols are used:

η local overvoltage (V)

η_(c) ^(+,0) positive cathode overvoltage at r=0 (V)

σ_(m) ionic conductivity of the membrane (Ω⁻¹ cm⁻¹)

Φ membrane potential (V)

Φ⁺ positive membrane potential (V)

Φ^(+,)− positive membrane potential for r→∞ (V)

ϕ potential of the carbon phase (V)

TABLE 1 Selected physical parameters for the calculations ORR exchangecurrent density j_(ox) ⁻⁰ A cm⁻² 10⁻⁶ ORR equilibrium potential E_(ox)^(eq) V 1.23 ORR Tafel slope of the half cell b_(ox) V 0.03 HOR exchangecurrent density in j_(hy) ⁰ A cm⁻² 1.0 the working area HOR Tafel slopeof the half cell b_(hy) V 0.015 Membrane layer thickness l_(m) cm 0.005(50 μm) Proton conductivity of the σ_(m) Ω⁻¹ cm⁻¹ 0.1 membrane Cellpotential ϕ_(c) V 0.82642 Average current density in the J A cm⁻² 1working area

1. Cell Model and Basic Equations

A PEMFC comprising concentric electrodes is considered, having ageometry and a coordinate system as illustrated in FIG. 6. Theelectrodes, which are a large-surface-area cathode and a small anode,are disposed on the two sides of the polymer electrolyte membrane havingthe layer thickness l_(m). A corresponding catalyst layer, and inparticular the anode catalyst layer (ACL) and the cathode catalyst layer(CCL), is generally (not shown in FIG. 7) present between the electrodesand the membrane.

The center of the concentric anode then corresponds to r=0, and r=R_(a)applies for the convex edge of the anode toward the recess. In thismodel, the cathode can likewise be regarded as a concentric electrodehaving a radius R_(c) where R_(c)→>∞. The region R_(a)<r≤R_(c) or ∞ isreferred to as an anode-free region (recess), and the region for0≤r≤R_(a) as the working area. This arrangement is representedschematically in FIG. 6.

A model was developed for distribution of the cathode overvoltage η_(c)and the membrane potential Φ in the anode-free region of the fuel cell.Mathematically, the problem results in the axially symmetricPoisson-Boltzmann equation for the cathode overvoltage η_(c). Thesolution to this problem demonstrates that the anode-free region |η_(c)|shows a rapid drop to zero as a function of the radius, while |Φ|quickly rises to the value of |η_(c) ⁰| of the cathode overvoltage inthe working area of the cell. The smaller the radius of the concentricanode R_(a), the faster the membrane potential Φ reaches the value ofthe cathode overvoltage in the working area η_(c) ⁰.

From this, it follows that, for measuring the cathodic overvoltagebetween the working electrodes (or the working and counter electrodes),the smaller the local radius R_(a) of this region, the closer thereference cell (RE) can be positioned to the convexly curved edge regionof the anode.

The case considered here is one in which the concentric cathode having aradius R_(c), serving as the working electrode, extends across theentire region of the membrane (endlessly), the anode however, serving asthe counter electrode, makes contact with only a portion of the surfaceof the membrane. The center of the anode corresponds to r=0. The anodehas a concentric geometry having the radius R_(a), wherein R_(a)<R_(c).This yields a kind of recess or anode-free region on the anode side onthe surface area of the electrolyte membrane with which the anode doesnot make contact.

It is also, of course, analogously possible to consider the case inwhich the anode represents the working electrode, and a concentriccathode is present as the counter electrode.

The goal initially is to understand the distribution of current and thepotentials in this system. The main variable in this problem is themembrane potential Φ, which follows the Poisson equation

$\begin{matrix}{{{\frac{1}{r}\frac{\partial\;}{\partial r}\left( {r\frac{\partial\Phi}{\partial r}} \right)} + \frac{\partial^{2}\Phi}{\partial z^{2}}} = 0} & (1)\end{matrix}$

An infinitely large cathode shall mean that the cathode radius is largerthan the membrane layer thickness l_(m) by several orders of magnitude.This assumption allows the second derivative along the z-axis inequation 3 to be replaced by the difference of the proton currentdensity into the membrane, and out of the same, resulting in thefollowing equation:

$\begin{matrix}{{\frac{1}{r}\frac{\partial\;}{\partial r}\left( {r\frac{\partial\Phi}{\partial r}} \right)} = \frac{j_{c} - j_{a}}{\sigma_{m}l_{m}}} & (2)\end{matrix}$

Here, j_(a) and j_(c) correspond to the current densities at the anodeside and the cathode side of the membrane. Moreover, it is assumed thatj_(a) and j_(c) follow Butler-Volmer kinetics.

$\begin{matrix}{j_{a} = {2j_{hy}{\sinh\left( \frac{\eta_{o}}{b_{hy}} \right)}\mspace{14mu}{and}}} & (1) \\{{j_{c} = {2j_{ox}{{\sinh\left( {- \frac{\eta_{c}}{b_{ox}}} \right)}.}}}\mspace{14mu}} & (2)\end{matrix}$

Here, j_(hy) and j_(ox) each denote the corresponding surface exchangecurrent densities of the anode catalyst layer (ACL) and of the cathodecatalyst layer (CCL), η_(a) and η_(c) are the corresponding localelectrode overvoltages at the anode and the cathode, and b_(hy) andb_(ox) are the Tafel slopes of the corresponding half-cell reactioncorresponding thereto. Since it is assumed that the transport losses aresmall, the dependence on the reactant concentration is already includedin j_(hy) and j_(ox).

The respective half-cell overvoltages follow fromη_(α)=ϕ_(α) −Φ−E _(HOR) ^(eq)  (3)η_(c)=ϕ_(c) −Φ−E _(ORR) ^(eq)  (4)wherein ϕ_(α) and ϕ_(c) represent the electrode potentials and E_(HOR)^(eq)=0 V and E_(ORR) ^(eq)=1.23 V represent the equilibrium potentialsof the corresponding half-cell reactions. It is assumed that the anodeis grounded (ϕ_(α))=0, and thus ϕ_(c) represents the cell potential.

Inserting equation (3) to equation (6) into equation (2) and introducingthe non-dimensional variables

$\begin{matrix}{{\overset{\sim}{r} = \frac{r}{l_{m}}},{\overset{\sim}{j} = \frac{{jl}_{m}}{\sigma_{m}b_{ox}}},{\overset{\sim}{\Phi} = \frac{\Phi}{b_{ox}}},{\overset{\sim}{\phi} = \frac{\phi}{b_{ox}}},{{\overset{\sim}{b}}_{hy} = \frac{b_{hy}}{b_{ox}}}} & (5)\end{matrix}$yields

$\begin{matrix}{{\frac{1}{\overset{\sim}{r}}\frac{d}{d\overset{\sim}{r}}\left( {\overset{\sim}{r}\frac{d\;\overset{\sim}{\Phi}}{d\overset{\sim}{r}}} \right)} = {{2{\overset{\sim}{j}}_{ox}^{0}{\sinh\left( {{- {\overset{\sim}{\phi}}_{c}} + \overset{\sim}{\Phi} + {\overset{\sim}{E}}_{ORR}^{eq}} \right)}} - {2{\overset{\sim}{j}}_{hy}^{0}{H\left( {{\overset{\sim}{R}}_{a} - \overset{\sim}{r}} \right)}{\sinh\left( {{- \overset{\sim}{\Phi}}\text{/}{\overset{\sim}{b}}_{hy}} \right)}}}} & (6)\end{matrix}$

Here, j _(ox) ⁰ including the superscript symbol ⁰ corresponds to thevalue at the center of the concentric working electrode (cathode). Hdenotes the Heaviside function, which in the working area assumes thevalue 1, and in the anode-free region assumes the value 0. The absenceof anodic catalyst outside the working area is taken into considerationby setting the exchange current density of the hydrogen oxidationreaction (HOR) to zero.

Within the scope of the invention, the distribution of the membranepotential Φ in the anode-free region is now examined. In this region,the current production at the anode side vanishes, and equation (8) issimplified to

$\begin{matrix}{{\frac{1}{\overset{\sim}{r}}\frac{d}{d\overset{\sim}{r}}\left( {\overset{\sim}{r}\frac{d\;\overset{\sim}{\Phi}}{d\overset{\sim}{r}}} \right)} = {2{\overset{\sim}{j}}_{ox}^{0}{\sinh\left( {{- {\overset{\sim}{\phi}}_{c}} + \overset{\sim}{\Phi} + {\overset{\sim}{E}}_{ORR}^{eq}} \right)}}} & (9)\end{matrix}$

It is now possible to represent this equation as a function of thecathode overvoltage according to equation (6). As a non-dimensionalequation, this results in{tilde over (η)}_(c)={tilde over (ϕ)}_(c) −{tilde over (Φ)}−{tilde over(E)} _(ORR) ^(eq)  (10)

Inserting this into equation (9) yields

$\begin{matrix}{{\frac{1}{\overset{\sim}{r}}\frac{d}{d\overset{\sim}{r}}\left( {\overset{\sim}{r}\frac{d\;{\overset{\sim}{\eta}}_{c}}{d\overset{\sim}{r}}} \right)} = {\kappa^{2}\sinh\;{\overset{\sim}{\;\eta}}_{c}}} & (11)\end{matrix}$where κ=√{square root over (2{tilde over (j)} _(ox) ⁰)}  (12)

By introducing the positive overvoltage,{tilde over (η)}_(c) ⁺=−{tilde over (η)}_(c)>0  (13)the following equation is obtained for {tilde over (η)}_(c) ⁺:

$\begin{matrix}{{\frac{1}{\overset{\sim}{r}}\frac{d}{d\overset{\sim}{r}}\left( {\overset{\sim}{r}\frac{d\;{\overset{\sim}{\eta}}_{c}^{+}}{d\overset{\sim}{r}}} \right)} = {\kappa^{2}\sinh\mspace{11mu}{\overset{\sim}{\eta}}_{c}^{+}}} & (14)\end{matrix}$

2. General Conditions

It was possible to demonstrate that, due to the very high exchangecurrent density of the hydrogen oxidation reaction (HOR), the cathodepotential {tilde over (η)}_(c) at the anode edge in polymer electrolytefuel cells is very close to the bulk potential value in the working areaaccording to equation 7. For a system having axial symmetry, it istherefore to be expected that {tilde over (η)}_(c) ⁺({tilde over(R)}_(α))≅η_(c) ⁺⁰, wherein {tilde over (η)}_(c) ⁺⁰≡{tilde over (η)}_(c)⁺(0)=−{tilde over (η)}_(c)(0) is the overvoltage of the oxygen reductionreaction (ORR) at the center of the axis of symmetry. Numerical testsconfirmed this assumption. As a result, however, the working area canadvantageously be eliminated from the consideration by placing thepotential {tilde over (η)}_(c) ⁺({tilde over (R)}_(α))=η_(c) ⁺⁰ at theedge of the anode. In this way, the general conditions for equation 14result as follows:

$\begin{matrix}{{\;{{{{\overset{\sim}{\eta}}_{c}^{+}\left( {\overset{\sim}{R}}_{a} \right)} = \;{\overset{\sim}{\eta}}_{c}^{+ 0}},\frac{d\;{\overset{\sim}{\eta}}_{c}^{+}}{d\overset{\sim}{r}}}}_{\overset{\sim}{r} = \infty} = 0} & (15)\end{matrix}$

The second equation means that the proton current density along {tildeover (r)} moves toward zero for ∞. The fact of an infinite cathodeexists approximately when the following condition is met: κ{tilde over(R)}_(c)>>1. The condition, however, is redundant. A more detailedanalysis shows that the approximation works well with an infinitecathode as long asR _(c)≥3L _(l,r)  (16)wherein L_(l,r) is given by equation 18. Such a condition can beregularly regarded as given for fuel cells on a laboratory scale.

3. Different Anodic Radii R_(a)

FIG. 7 shows the solutions for equation 14 for different anode radiiR_(a). The particular characteristic of this problem is that thegradient of the potential of the oxygen reduction reaction (ORR) η_(c) ⁺increases very close to the anode edge as the anode radius decreases.Qualitatively, this resembles the behavior of Laplace's potentialbetween a charged metal tip and a plane: the narrower the radius of themetallic tip, the more strongly the potential drops in the vicinity ofthe tip as a function of the axial symmetry.

The practical significance that can be derived from FIG. 7 is that, in afuel cell arrangement comprising a large-surface-area cathode and aconcentric anode, the smaller the radius of the anode, the closer thereference electrode (RE) can be disposed to the anode. This appliesunder the prerequisite that this arrangement is to be used, and can beused, to ascertain the cathodic overvoltage with sufficient accuracy.

4. Position of the Reference Electrode

It becomes apparent from FIG. 8 that the positive membrane potential Φ⁺approaches the limiting value η_(c) ^(+,0) more quickly as the anoderadius R_(a) decreases. Likewise, the radial width of the region {tildeover (L)}_(l,r) dominated by the oxygen reduction reaction is apparentfrom FIG. 8. This radial width increases with the increasing anoderadius R_(a).

For further estimations, the assumption is made that the referenceelectrode (RE) is disposed at a distance {tilde over (L)}_(l,r) from theat least partially convexly curved anode edge having the local radiusR_(a). FIG. 8 shows that this assumption results in the determination ofη_(c) ⁺ with a 10% accuracy. It is advisable to compare this distance{tilde over (L)}_(l,r) (FIG. 9, solid line) to the analogous distance{tilde over (L)}_(l,r) for a straight anode edge according to FIG. 3.The dotted line in FIG. 9 represents the value of {tilde over(L)}_(l,r). Physically, {tilde over (L)}_(l,r) is an asymptote,approached by the {tilde over (L)}_(l,r) curve for {tilde over(R)}_(α)→∞ (FIG. 9). For very small radii of the anode, which is to sayfor {tilde over (R)}_(α)≥10, the radial distance {tilde over (L)}_(l,r)between the curved anode edge and the reference electrode is only halfas small as the distance

${\overset{\sim}{L}}_{1,x} \cong \frac{1,4}{\kappa}$for the corresponding anode geometry having a straight edge, as shown inFIG. 4. But even at larger radii, for example for {tilde over(R)}_(α)≈100 (right edge in FIG. 9), the radial distance {tilde over(L)}_(l,r) between the curved anode edge and the reference electrode isstill almost 30% smaller than the distance {tilde over (L)}_(l,r) thatwould result for the corresponding anode geometry having a straightedge.

It must be noted that the distance {tilde over (L)}_(l,r) decreasesdrastically for very small anode radii. For the range of {tilde over(R)}_(α)≤1, it is questionable whether the underlying model can still beapplied, since possible strong two-dimensional effects in closeproximity to the concavely curved anode edge are ignored. Thus, for theoptimal minimum distance, {tilde over (R)}_(α) would advantageously beselected in the range of 2 to 3.

A possible expression that describes the dependencies of {tilde over(L)}_(l,r) as a function of {tilde over (R)}_(α) in the region of0≤η{tilde over (R)}_(α)≤1 is given by

$\begin{matrix}{{\overset{\sim}{L}}_{1,r} \cong {{\frac{\pi}{2\kappa}\left\lbrack {\ln\left( \frac{67}{18\left( {\kappa\;{\overset{\sim}{R}}_{a}} \right)^{7/45}} \right)} \right\rbrack}^{- 1}\mspace{14mu}{for}\mspace{14mu} 0} \leq {\kappa\;{\overset{\sim}{R}}_{a}} \leq 1} & (17)\end{matrix}$and is represented in FIG. 9 by open circles. In the non-dimensionalcase, this yields the following equation:

$\begin{matrix}{{L_{1,r} \cong {\frac{{\pi\lambda}_{D}}{2}\left\lbrack {\ln\left( \frac{67}{18\left( \;{R_{a}\text{/}\lambda_{D}} \right)^{7/45}} \right)} \right\rbrack}^{- 1}},{{{where}\mspace{14mu}\lambda_{D}} = \sqrt{\frac{\sigma_{m}b_{ox}l_{m}}{2j_{ox}^{0}}}}} & (18)\end{matrix}$

This equation 18 can advantageously be used to estimate the requiredminimum distance L_(gap) between the anode tip (convexly curved edgeregion) having the local radius R_(a) and the reference electrode in anapplication in terms of development. If a certain the measurementaccuracy is required, L_(gap)≅3·L_(l,r) should be used as the minimumdistance.

FIGS. 4 and 5 show possible arrangements for a fuel cell, eachcomprising a reference electrode. FIG. 3 shows the arrangement of areference cell in a fuel cell having a straight anode edge according tothe prior art. In this case, the minimum distance results as

${L_{gap} \cong {3 \cdot \lambda_{D}}},{{{where}\mspace{14mu}\lambda_{D}} = {\sqrt{\frac{\sigma_{m}b_{ox}l_{m}}{2j_{ox}^{0}}}.}}$

FIG. 5, in contrast, shows the case for an embodiment of the arrangementaccording to the invention in which the anode comprises a convexlycurved edge region in close proximity to the reference electrode. Here,L_(gap)≅3·L_(l,r) applies for the minimum distance, wherein L_(l,r) isgiven by equation 18. It should be noted that L_(gap) is smaller in FIG.5 than in FIG. 3. FIG. 5 shows a fuel cell arrangement comprising ananode edge having a convexly curved tip. This tip causes the membranepotential Φ⁺ to increase drastically and quickly as the distance awayfrom the tip increases. In this case, the reference electrode can thusbe disposed very close to the tip, without interfering with the accuracyof the measurement. The value of the distance L_(gap) in FIG. 5 can beeasily determined by equation 18 by using the local radius of the anodetip for R_(a).

Using the parameters from Table 1 and an anode radius R_(a)=0.01 cm, adistance L_(l,r)=1.97 cm would be obtained from equation 18. For anarrangement comprising an appropriately straight anode edge, the samemembrane potential would not be achieved until a distance ofL_(l,x)=3.83 cm. At smaller anode radii R_(a), the advantage would beeven greater, due to the curved anode edge.

The invention claimed is:
 1. A method for ascertaining the overvoltageof a working electrode in a fuel cell, comprising a polymer electrolytemembrane, the working electrode, a counter electrode, and a referenceelectrode, and in which the working electrode is a continuous electrodeand is disposed on one side of the polymer electrolyte membrane and thecounter electrode is grounded and is disposed on the other side of thepolymer electrolyte membrane, the method comprising: measuring andcomparing a potential of the reference electrode and a potential of thecounter electrode in the fuel cell configured in which the counterelectrode has an outer edge spanning an entire circumference of thecounter electrode along a surface of the polymer electrolyte membrane,said outer edge comprising a first portion forming a lateral edge havinga convex curvature with a local radius R_(a) and said outer edgecomprising a second portion with a local radius greater than R_(a), sothat said lateral edge forms a convexly curved tip which is moreconvexly curved than said second portion, the polymer electrolytemembrane surface comprises an electrode-free region adjoining thecounter electrode and opposite the working electrode, and the referenceelectrode is disposed on the polymer electrolyte membrane surface in theregion of the electrode-free region and in the immediate vicinity of theconvex edge region of the counter electrode at a distance L_(gap),wherein the minimum distance for L_(gap) between the reference electrodeand the convexly curved edge region of the counter electrode in theregion is given by L_(gap) = 3 ⋅ L_(l, r)  where$L_{1,r} \cong {{\frac{{\pi\lambda}_{D}}{2}\left\lbrack {\ln\left( \frac{67}{18\left( {R_{a}\text{/}\lambda_{D}} \right)^{7/45}} \right)} \right\rbrack}^{- 1}\mspace{14mu}{and}}$$\mspace{11mu}{\lambda_{D} = \sqrt{\frac{\sigma_{m}b_{ox}l_{m}}{2j_{ox}^{0}}}}$where σ_(m)=ionic conductivity of the polymer electrolyte membrane (Ω⁻¹cm⁻¹), b_(ox)=Tafel slope of the half cell for the electrochemicalreaction of the working electrode (V), l_(m)=polymer electrolytemembrane layer thickness (cm), j_(ox) ⁰=exchange current density of thecatalyst of the working electrode per unit of electrode surface in (Acm⁻²), and R_(a)=radius of the convexly curved edge region of thecounter electrode (cm).
 2. The method according to claim 1, wherein thedistance between the reference electrode and the convexly curved edgeregion of the counter electrode is selected in the range between L_(gap)and 3 L_(gap), from the convexly curved edge region of the counterelectrode.
 3. The method according to claim 1, wherein the referenceelectrode is supplied with hydrogen during the method.
 4. A methodaccording to claim 1, wherein an anode is used as the counter electrode,and a cathode is used as the working electrode.
 5. A method according toclaim 1, wherein a value between 0.01 and 1 cm is selected for the localradius of curvature of the curved edge region of the counter electrode.6. A method according to claim 1, wherein the fuel cell is one of apolymer electrolyte membrane fuel cell (PEMFC), a direct methanol fuelcell (DMFC), a solid oxide fuel cell (SOFC), and a high-temperaturepolymer electrolyte membrane fuel cell (HT-PEMFC).
 7. The methodaccording to claim 1, wherein the counter electrode disposed on saidother side of the polymer electrolyte membrane has an outer edgespanning an entire outer periphery along a surface of polymerelectrolyte membrane, and wherein a shortest distance from a lateraledge of the working electrode to a nearest edge along said outer edge ofthe counter electrode is prescribed so that said nearest edge of thecounter electrode does not influence the overvoltage of the workingelectrode in the working area of the fuel cell.
 8. The method accordingto claim 1, wherein the distance between the reference electrode and theconvexly curved edge region of the counter electrode is selected in therange of Lgap to 100 Lgap from the convexly curved edge region of thecounter electrode.
 9. The method according to claim 1, wherein thedistance between the reference electrode and the convexly curved edgeregion of the counter electrode is selected in the range between Lgapand 10 Lgap from the convexly curved edge region of the counterelectrode.
 10. A method according to claim 1, wherein a value of lessthan 0.1 cm is selected for the local radius of curvature of the curvededge region of the counter electrode.
 11. A method for ascertaining theovervoltage of a working electrode in a fuel cell comprising a polymerelectrolyte membrane, the working electrode, a counter electrode, and areference electrode, and in which the working electrode is a continuouselectrode and is disposed on one side of the polymer electrolytemembrane and the counter electrode is grounded and is disposed on theother side of the polymer electrolyte membrane, the method comprising:determining a minimum distance to be configured between the referenceelectrode and the counter electrode; configuring the fuel cell to havesaid polymer electrolyte membrane with said working electrode disposedas a continuous electrode on one face of the polymer electrolytemembrane and to have the counter electrode and the reference electrodedisposed on an opposite face of the polymer electrolyte membrane, thecounter electrode and the reference electrode being spaced by at leastsaid minimum distance, the counter electrode being grounded, the polymerelectrolyte membrane surface comprising an electrode-free regionadjoining the counter electrode and opposite the working electrode; andmeasuring and comparing a potential of the reference electrode and apotential of the counter electrode in the configured fuel cell; andwherein the counter electrode has an outer edge spanning an entire outercircumference of the counter electrode along a surface of the polymerelectrolyte membrane, said outer edge comprising a first portion forminga lateral edge having a convex curvature with a local radius R_(a) andsaid outer edge comprising a second portion with a local radius greaterthan R_(a), so that said lateral edge forms a convexly curved tip whichis more convexly curved than said second portion, said minimum distancebeing from a distal tip of said convexly curved tip through saidelectrode-free region to a nearest edge of said reference electrode;wherein said minimum distance is determined as beingL_(gap) = 3 ⋅ L_(l, r)  where$L_{1,r} \cong {{\frac{{\pi\lambda}_{D}}{2}\left\lbrack {\ln\left( \frac{67}{18\left( {R_{a}\text{/}\lambda_{D}} \right)^{7/45}} \right)} \right\rbrack}^{- 1}\mspace{14mu}{and}}$$\mspace{11mu}{\lambda_{D} = \sqrt{\frac{\sigma_{m}b_{ox}l_{m}}{2j_{ox}^{0}}}}$where σ_(m)=ionic conductivity of the polymer electrolyte membrane (Ω⁻¹cm⁻¹), b_(ox)=Tafel slope of the half cell for the electrochemicalreaction of the working electrode (V), l_(m)=polymer electrolytemembrane layer thickness (cm), f_(ox) ⁰=exchange current density of thecatalyst of the working electrode per unit of electrode surface in (Acm⁻²), and R_(a)=radius of the convexly curved edge region of thecounter electrode (cm).
 12. A method according to claim 11, wherein ananode is used as the counter electrode, and a cathode is used as theworking electrode.
 13. A method according to claim 11, wherein the fuelcell is one of a polymer electrolyte membrane fuel cell (PEMFC), adirect methanol fuel cell (DMFC), a solid oxide fuel cell (SOFC), and ahigh-temperature polymer electrolyte membrane fuel cell (HT-PEMFC).